cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A307637 Sum of the second largest parts of the partitions of n into 7 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 6, 8, 14, 14, 16, 23, 28, 35, 38, 45, 52, 71, 66, 85, 94, 115, 121, 163, 154, 212, 194, 260, 253, 344, 289, 411, 382, 516, 457, 640, 533, 786, 652, 914, 778, 1112, 857, 1299, 1048, 1501, 1195, 1780, 1345
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A010051, A259197, A308974, A308975, A308976, A308977, A308978, A308979, A308980.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{7}],AllTrue[#,PrimeQ]&][[All,2]]],{n,0,60}] (* Harvey P. Dale, Oct 23 2022 *)

Formula

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} c(i) * c(j) * c(k) * c(l) * c(m) * c(o) * c(n-i-j-k-l-m-o) * i, where c = A010051.
a(n) = A308974(n) - A308975(n) - A308976(n) - A308977(n) - A308978(n) - A308979(n) - A308980(n).

A308975 Sum of the smallest parts of the partitions of n into 7 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 8, 9, 8, 13, 12, 18, 16, 20, 20, 32, 24, 37, 32, 45, 38, 63, 44, 74, 52, 84, 62, 109, 66, 123, 84, 145, 94, 173, 102, 209, 120, 225, 136, 272, 146, 309, 172, 343, 190, 405, 206, 466, 232, 499, 262
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A010051, A259197, A307637, A308974, A308976, A308977, A308978, A308979, A308980.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[o*(PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[n - i - j - k - l - m - o] - PrimePi[n - i - j - k - l - m - o - 1]), {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

Formula

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} c(i) * c(j) * c(k) * c(l) * c(m) * c(o) * c(n-i-j-k-l-m-o) * o, where c = A010051.
a(n) = A308974(n) - A308976(n) - A308977(n) - A308978(n) - A308979(n) - A307637(n) - A308980(n).

A308976 Sum of the sixth largest parts of the partitions of n into 7 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 9, 9, 9, 13, 14, 18, 18, 20, 24, 32, 29, 37, 41, 47, 48, 65, 57, 76, 69, 88, 85, 115, 90, 129, 120, 157, 132, 187, 150, 225, 176, 247, 202, 298, 221, 339, 266, 385, 293, 453, 328, 522, 366, 565, 426
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A010051, A259197, A307637, A308974, A308975, A308977, A308978, A308979, A308980.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[m*(PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[n - i - j - k - l - m - o] - PrimePi[n - i - j - k - l - m - o - 1]), {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

Formula

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} c(i) * c(j) * c(k) * c(l) * c(m) * c(o) * c(n-i-j-k-l-m-o) * m, where c = A010051.
a(n) = A308974(n) - A308975(n) - A308977(n) - A308978(n) - A308979(n) - A307637(n) - A308980(n).

A308977 Sum of the fifth largest parts of the partitions of n into 7 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 7, 9, 10, 9, 15, 14, 20, 18, 24, 24, 39, 31, 47, 43, 58, 50, 81, 61, 97, 75, 116, 93, 146, 100, 172, 134, 207, 148, 246, 170, 298, 202, 332, 232, 395, 257, 463, 314, 521, 343, 612, 392, 720, 452, 788
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A010051, A259197, A307637, A308974, A308975, A308976, A308978, A308979, A308980.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[l*(PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[n - i - j - k - l - m - o] - PrimePi[n - i - j - k - l - m - o - 1]), {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

Formula

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} c(i) * c(j) * c(k) * c(l) * c(m) * c(o) * c(n-i-j-k-l-m-o) * l, where c = A010051.
a(n) = A308974(n) - A308975(n) - A308976(n) - A308978(n) - A308979(n) - A307637(n) - A308980(n).

A308978 Sum of the fourth largest parts of the partitions of n into 7 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 5, 7, 10, 10, 11, 15, 16, 20, 24, 26, 32, 43, 40, 51, 55, 64, 70, 93, 81, 111, 102, 132, 128, 172, 139, 202, 182, 243, 209, 296, 233, 352, 287, 402, 336, 495, 372, 577, 458, 661, 520, 800, 585, 938, 683
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A010051, A259197, A307637, A308974, A308975, A308976, A308977, A308979, A308980.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[k*(PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[n - i - j - k - l - m - o] - PrimePi[n - i - j - k - l - m - o - 1]), {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

Formula

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} c(i) * c(j) * c(k) * c(l) * c(m) * c(o) * c(n-i-j-k-l-m-o) * k, where c = A010051.
a(n) = A308974(n) - A308975(n) - A308976(n) - A308977(n) - A308979(n) - A307637(n) - A308980(n).

A308979 Sum of the third largest parts in the partitions of n into 7 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 5, 5, 8, 10, 12, 11, 19, 18, 27, 28, 35, 36, 57, 48, 67, 67, 87, 82, 121, 99, 146, 126, 176, 156, 232, 181, 271, 238, 336, 277, 414, 325, 500, 405, 588, 480, 722, 542, 843, 660, 977, 752, 1172, 851, 1374
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A010051, A259197, A307637, A308974, A308975, A308976, A308977, A308978, A308980.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[j*(PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[n - i - j - k - l - m - o] - PrimePi[n - i - j - k - l - m - o - 1]), {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

Formula

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} c(i) * c(j) * c(k) * c(l) * c(m) * c(o) * c(n-i-j-k-l-m-o) * j, where c = A010051.
a(n) = A308974(n) - A308975(n) - A308976(n) - A308977(n) - A308978(n) - A307637(n) - A308980(n).

A308980 Sum of the largest parts in the partitions of n into 7 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 3, 8, 8, 15, 20, 20, 24, 40, 42, 62, 66, 73, 92, 132, 122, 172, 180, 211, 237, 324, 296, 394, 370, 470, 463, 645, 521, 756, 708, 916, 845, 1146, 935, 1403, 1158, 1576, 1372, 1953, 1547, 2330, 1898, 2623, 2217
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A010051, A259197, A307637, A308974, A308975, A308976, A308977, A308978, A308979.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[(n-i-j-k-l-m-o)*(PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[n - i - j - k - l - m - o] - PrimePi[n - i - j - k - l - m - o - 1]), {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

Formula

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} c(i) * c(j) * c(k) * c(l) * c(m) * c(o) * c(n-i-j-k-l-m-o) * (n-i-j-k-l-m-o), where c = A010051.
a(n) = A308974(n) - A308975(n) - A308976(n) - A308977(n) - A308978(n) - A308979(n) - A307637(n).
Showing 1-7 of 7 results.

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